Asymptotic of ‘rigid-body’ motions for nonlinear dynamics:
physical insight and methodologies 1
V.N. Pilipchuk
Wayne State University, Detroit, Michigan 48202
Abstract: The purpose of the present work is to show that an adequate basis for understanding the essentially nonlinear phenomena must also be essentially nonlinear
however still simple enough to play the role of a basis. It is shown that such types of
‘elementary’ nonlinear models can be revealed by tracking the hidden links between
analytical tools of analyses and subgroups of the rigid-body motions or, in other
terms, rigid Euclidean transformation. While the subgroup of rotations is linked with
linear and weakly nonlinear vibrations, the translations with reflections can be
viewed as a geometrical core of the strongly nonlinear dynamics associated with the
so-called vibro-impact behaviors. It is shown that the corresponding analytical approach develops through non-smooth temporal substitutions generated by the impact
models.
1. Introduction
This work is motivated by the intent to introduce a unified physical basis for analyzing vibrations of essentially unharmonic, non-smooth or may be discontinuous
time shapes. Transitions to non-smooth limits can make investigations especially
difficult due to the fact that the dynamic methods were originally developed
within the paradigm of smooth motions based on the classical theory of differential equations. From the physical standpoint, such way is natural for modeling the
low-energy motions. Although the impact dynamics has also quite a long prehistory, non-smooth behaviors are often viewed as an exemption rather than a rule.
Notice that the classical theory of differential equations usually avoids nondifferentiable and discontinuous functions. Presently, however, many theoretical
and applied areas are dealing with the high-energy phenomena accompanied by
strongly non-linear spatio-temporal behaviors making the classical smooth methods difficult to apply. For instance, such phenomena are considered in engineering analyses of dynamical systems under constraint conditions, friction-induced
vibrations, structural damages due to cracks, liquid sloshing impacts, etc. Similarly to the well-known analogy between mechanical and electrical harmonic oscillators, the so-called Schmitt trigger circuits generate non-smooth signals whose
temporal shapes resemble the mechanical vibro-impact processes. In many such
cases, it is still possible to adapt different smooth methods of the dynamic analyses through strongly non-linear algebraic manipulations with state vectors or by
splitting the phase space into multiple ‘smooth domains.’ As a result, formulations
are often reduced to the discrete mappings in a wide range of the dynamics from
1 A keynote lecture at 12th DSTA Conference, December 2-5, 2013 Łódź, Poland
2
periodic to stochastic. It will be shown that a complementary analytical tool can be
built on generating models developing essentially unharmonic behaviors as their
inherent properties. For instance, the methodology presented in this work, employs elementary impact systems as a physical basis for describing different types
of unharmonic processes. This is implemented through the non-smooth time substitutions introduced originally for strongly nonlinear however smooth models [1].
Besides, this tool reveals explicit links between the impact dynamics and the algebra of hyperbolic numbers analogously to the link between harmonic vibrations
and conventional (elliptic) complex analyses [2].
2. Physical and mathematical principles
2.1. Linear and elementary nonlinear phenomena
Although the notion of linearity is quite clear in terms of mathematical formulations, the attempt to directly associate the mathematical definition with physical
phenomena faces ambiguities due to the fact that the differential equations of motion for the same model may appear to be either linear or nonlinear if switching
between different types of coordinates. For instance, a mechanical system, which
is linear in Cartesian coordinates, becomes nonlinear in polar coordinates. Nevertheless, recognizing the unique role of Cartesian coordinates, it was suggested to
define a mechanical system as linear if the corresponding differential equations of
motion are linear in Cartesian coordinates 2 . The purpose of this subsection is to
determine the most elementary dynamic phenomena that can already be qualified
as essentially nonlinear. For that reason, it is convenient to consider the linear
situation first, for instance, on a typical mass-spring vibrating model; see figure 1
(a), where all the springs are assumed to be linearly elastic. The corresponding
position vector is represented in the following complex form
q = A1 exp(iω1t + ϕ1 ) + A2 exp(iω 2 t + ϕ 2 )
where Ak are constant complex vectors,
ωk
and
ϕk
(1)
(k = 1,2) are the modal fre-
quencies and initial phases.
The fragment (b) in figure 1 shows another mechanical system whose dynamics is described by the same expression (1), if the system’ position is identified by
the two points fixed at the edges of the discs. Therefore, the one-dimensional dynamics generated by linearly elastic restoring forces can be represented by the
rigid-body rotations. In other words, the linearly elastic forces are effectively
eliminated by increasing the dimension of space. To some extent, such simple
observation provides an explanation why the sine and cosine waves possess so
convenient mathematical properties. Namely these harmonic functions represent
2
This definition was suggested by V. Zhuravlev (private communication, Moscow, 1989.)
3
the subgroup of elementary rigid-body motions, such as rotation. Furthermore, the
link between the two models, (a) and (b), in figure (1), enables one to associate the
linearity with the subgroup of rotations.
ω2
q = {u1 , u1}
ω1
u2
u1
A1
(a)
A2
(b)
reflection
translation
t=a
t
perfectly stiff plane (mirror)
(c)
(d)
Figure 1. Vibration of the linearly elastic system (a) is represented by the rigid-body rotations (b); the
impact oscillator (c) is associated with the rigid-body translation and reflection (d).
Mechanical model
t=a
Basic function
Substitution
s =| t − a |
t = a + ss
v =1
1
s(t)
1
1
v =1
2
τ=
τ(t)
2
π
 πt 
arcsin sin 
2
v =1
x=0
s(t;d)
t = 1 + (τ − 1)τ
τ 2 = 1
∞
x
3
s 2 = 1
x =1
s (t ; d )
t =  (ti + si ) si
i =0
s 2 = 1
1
= ( d + | t | − | t − d |) i
2
Figure 2. Different versions of non-smooth temporal substitutions generated by the impact models.
4
Apparently, the rotations do not cover all the rigid-body motions – rigid
Euclidean transformations. These include also translations and reflections. The
translation itself seems to have a little physical content. However, combinations
of translation and reflection appear to be more interesting as shown in figure 1 (c)
and (d) through the corresponding mechanical representations. In particular, the
fragment (c) shows a typical impact oscillator, whereas the fragment (d) illustrates
a single reflection case, which can be viewed as the most elementary nonlinear
phenomenon.
2.2. Impact models and non-smooth temporal substitutions
Based on the class of linear generating models, whose typical representative is
shown in figure 1 (a), the quasi-harmonic methods for nonlinear vibration theory
are well developed and widely used. In contrast, using the models associated with
the translation-reflection, as those shown in figure 1 (c-d), is less common. Despite of being geometrically simple, such models are not described within the classical theory of differential equations due to the presence of discontinuities in the
dynamic states. This fact essentially complicates any direct use of the impact
models as a basis for building asymptotic or iterative procedures. It is shown,
however, that appropriate preliminary adaptations of the differential equations of
motion can be conducted through the non-smooth time substitutions listed in the
third column of figure 2. In particular, the first row explains how such type of
substitutions is introduced. Namely, the basic function s =| t − a | is the model’
coordinate, which can be interpreted as an eigentime of the system provided that
nothing else happens except the translation and reflection under consideration.
The goal is to introduce a new temporal argument, say s, in order to obtain the differential equation of motion for the model, which is shown on the left of first row
in figure 2. Apparently, the substitution t → s cannot work directly since no inversion t = t (s ) does exist on the entire time domain. Nevertheless, the following
generalization holds t = t ( s, s) ; see the first row on its right in figure 2 for details.
In particular, this inversion appears to have the specific algebraic structure of hyperbolic numbers with the basis {1, s} . In contrast to the conventional elliptic
complex numbers, the unipotent s , if squared, gives the positive sign
2
namely s = 1 . Interestingly enough, the hyperbolic numbers have been known
for about one and a half century as abstract algebraic elements with no relation to
the nonlinear dynamics or non-smooth functions. Moreover, such numbers have
been re-introduced multiple times under different names; see [3, 4] for an explanation of this fact and further details. In the present case, the hyperbolic structures
are generated by the non-smooth temporal arguments of impact systems, while the
unipotent s possesses a clear physical interpretation as the normalized velocity of
impact system. Different applications to the vibration problems are based on the
periodic version [1, 2] illustrated by the second row of figure 2. In particular, the
5
equation on the right of the row means that, during one period, the time argument
is expressed through the dynamic states of impact oscillator in the form of hyperbolic number with the unipotent τ . This statement therefore applies to any periodic process as was originally shown in [1]. Figure 3 provides a geometrical interpretation of the ‘oscillating time’ τ for the particular case of periodic process
described by the even function x(t ) with respect to the quarter of period, t = a .
Finally, in the third row of figure 2, the original time argument is structurised
to match the one-dimensional dynamics of rigid-body chain of identical particles.
The continuous ‘global’ time is associated with the propagation of linear momentum, whereas a sequence of non-smooth ‘local’ time arguments describes the behavior of individual physical particles. Such an idea helps to incorporate the temporal symmetries of the dynamics into the differential equations of motion in
many other cases of regular or irregular sequences of internal impacts or external
pulses. Since the local times are bounded, a wider range of analytical tools becomes applicable.
x
A
x
-a
aτ
a
t
t
-A
Figure 3. The ‘oscillating time’ τ changes its direction whenever a system makes a U – turn, while the
original physical time t runs to infinity.
2.3. Different asymptotic approaches to the vibration theory
Figure 4 provides further illustration for logical links between two alternative approaches to the vibration problems. Interestingly enough, the illustration is possible within the same one-degree-of-freedom model, which is shown at the first row
of figure 4. Note that the oscillators with power-form characteristics were considered for quite a long time. For instance, Lyapunov obtained such oscillators while
investigating degenerated cases of the dynamic stability problems. Besides, he introduced a couple of special functions, cs and sn, in order to invert the corresponding quadratures [5]. This relatively simple model nevertheless depicts the gradual
transition from linear to strongly nonlinear dynamics as the exponent n runs from
unity to infinity. Notably, all the temporal mode shapes of the oscillator are described by the special functions cs and sn, except the two boundaries of the inter-
6
val 1 ≤ n < ∞ . Both boundaries represent simple asymptotic limits described
within the class of elementary functions.
Impact limit
Harmonic limit
1
x + x 2 n−1 = 0, n → ∞
x + x 2 n−1 = 0, n = 1
Impact oscillator
Harmonic oscillator
1
2
1
sin t
0
−1
t
2π
π
cos t
t
0
−1
Sine & cosine waves
Subgroup of rotations
z = x + ix
3
τ (t )
τ(t )
x = X (τ ) + Y (τ )τ
i = −1
τ 2 = 1
Elliptic complex numbers
yi
Hyperbolic complex numbers
Y τ
z
ϕ
4
Triangular sine wave
Translation-reflection
2
i
2
τ
1
x
1
4
X
| x |H = 1
i−
| x |H = 0
| z |= 1
| x | 2H = X
| z |2 = x 2 + y 2
A
k
5
cos kt + Bk sin kt
k =0
Fourier series
−Y
2
Hyperbolic complex plane
Elliptic complex plane
∞
2
∞
1
 k! X
k =0
(k )
(0)τ k +
1 (k )
Y (0)τ kτ
k!
Saw-tooth power series
Figure 4. Two alternative approaches to the vibration theory based on the harmonic and impact limits.
7
Consider first the limit of harmonic oscillator ( n = 1 ), generating the sine and
cosine waves; see the left of the second row. The widely known convenience of
using this couple of functions can be explained by their links to the elementary
rigid-body motions namely the subgroup of rotations. The algebra of conventional
(elliptic) complex numbers with the corresponding complex plane representation
can be viewed as a next hierarchic step here due to the well-known Euler formula.
Finally, taking the linear combination of harmonic waves with different frequencies and keeping in mind the idea of parameter variations leads the area of harmonic and quasi harmonic analyses of vibrating systems. Such tools therefore
represent the dynamic processes as a combination of the elementary rigid-body rotations with different angular speeds.
Let us consider now the limit n → ∞ , when the restoring force vanishes inside
the interval − 1 < x < 1 but becomes infinitely large as the system reaches the
potential barriers at x = ±1 . The physical meaning of this limit is introduced at
the top of the right column in figure 4. Despite of the strong (impact) nonlinearity,
the limiting oscillator is also described by quite simple elementary functions such
as the triangular sine and rectangular cosine, say τ and τ . These two non-smooth
functions are associated with another subgroup of the rigid-body motions namely
translation and reflection. Therefore, analogously to the case n = 1 , the upper
limit n = ∞ can play the same fundamental role by generating a hierarchy of tools
as listed in the right column of figure 4.
3. Further mathematical properties and examples
This section describes the basic mathematical properties of the non-smooth temporal substitutions introduced in figure 2. These properties are used then for derivations in different illustrating examples.
3.1. Mathematical properties
Consider first, the single reflection case; see the first row of figure 2. Algebraic,
differential, and integral properties are as follows:

Isomorphism with 2×2 symmetric matrixes
t 2 = (a + ss) 2 = a 2 + s 2 + 2ass
2
 a s   a2 + s2
 = 
tˆ = 
 s a   2as
2

2as 

a + s 2 
2
Functional linearity holds for any function x(t):
8
x(t ) = x(a + ss) = X ( s) + Y ( s) s
(2)
1
X ( s ) = [ x(a + s ) + x(a − s )]
2
1
Y ( s ) = [ x(a + s ) − x(a − s )]
2
For instance, exp(t ) = exp(a)(cosh s + s sinh s )

Division is conditioned to exclude the possibility of zero denominators:
t −1 =
(a − ss)
a − ss
a
s
= 2 2 = 2 2 − 2 2 s
(a + ss)(a − ss) a − s
a −s
a −s
(s ≠ a )

Sequential differentiation remains in the algebra of hyperbolic numbers
under the smoothness conditions at s = 0:
d
[ X ( s ) + Y ( s ) s] = Y ′( s ) + X ′( s ) s
dt
if Y (0) = 0
d2
[ X ( s ) + Y ( s ) s] = X ′′( s ) + Y ′′( s ) s
dt 2
if Y (0) = 0, X ′(0) = 0

Integration remains in the algebra of hyperbolic numbers:
 [ X (s(t )) + Y (s(t ))s(t )]dt
 s

s
=   Y ( z )dz + C  +   X ( z )dz  s
 0

0
(3)
9
3.2. Sample solution procedure for the s-case
Let us consider the following initial value problem
x + λx = 2 pδ (t − a ) ≡ ps
x ( 0) = 0
s =| t − a |
(4)
Introducing the new temporal argument, t → s , generates the following substitution for the unknown function: x(t ) = X ( s ) + Y ( s ) s . As a result, the differential equation takes the form
′+
Y
X + ( X ′ + λY ) s + (Y − p )s = 0
λ


 


regular
step− wise discont.
(5)
singular term
Equating separately the terms of different level of singularity in (5) to zero
leads to the following boundary value problem with no singular terms
Y ′ + λX = 0

 X ′ + λY = 0
 p − Y (0) = 0

(6)
The initial condition in (4) yields
X ( a) − Y (a ) = 0
(7)
The boundary value problem (6)-(7) is easy to solve in few steps. Then the
corresponding solution of the original initial value problem (4) is obtained in the
closed-form x(t ) = p exp(−λs )(1 + s) ; see figure 5 for illustration.
3
2
1
0
-1
0
1
2
3
4
5
t
Figure 5. Solution of problem (4) under the following parameters: P = 1.0; a = 1.0; λ = 1.0.
10
3.3. Properties and sample solution for the τ-case
Algebraic and differential properties of the periodic τ-version, which is shown in
the second row of figure 2, are similar to those listed in section 3.1. Note that, although the analytical definitions for the basic functions τ and τ look more complicated in this case, there is no need for memorizing them. All what is necessary
for solving problems is the following properties
τ = e, e 2 = 1, e ≠ 0 ⇔ τ = ±1
(8)
The third relationship in (8) means that, whenever τ reaches its amplitude values, a δ-spike occurs from the following series
∞
e(t ) = 2  [δ (t + 1 − 4k ) − δ (t − 1 − 4k )]
k = −∞
In other words, the present situation is quite similar to analytical manipulations with the conventional trigonometric functions using only the function properties with no involvement of their definitions. In order to illustrate the manipulations, let us consider the overdamped oscillator under the rectangular cosine
loading
x + λx = pe(t )
(9)
The unknown periodic solution is represented in the form [1, 2]
x = X (τ ) + Y (τ )τ
(10)
Substituting (10) in (9) and taking into account properties (8) gives
′+
Y
λ
X + ( X ′ + λY − p)e + Ye = 0




regular
step − wise discont.
(11)
singular
Equating separately to zero the terms of different level of singularity gives the
autonomous boundary value problem with no discontinuities
 X ′ + λY = p

Y ′ + λX = 0
Y (±1) = 0

(12)
11
Substituting solution of the boundary value problem (12) in (10) gives finally
the periodic closed form solution of the original equation (9); see figure 6 for illustration:
x=
p  sinh(λτ )  cosh(λτ )  
+ 1−
e

λ  cosh λ 
cosh λ  
(13)
1.0
0.5
0.0
- 0.5
- 1.0
0
2
4
6
8
t
Figure 6. Solution (13) for two different magnitudes of the parameter: λ = 1.0 – solid line, larger amplitude; λ = 5.0 – thin line, smaller amplitude.
Obviously, equation (9) can be also solved by means of either Fourier series or
Laplace transforms, or directly by matching different pieces of solution under periodicity conditions. However, using the Fourier series, for instance, requires a
very large number of terms near the non-smoothness points as shown in figure 6.
4. Conclusions
This paper outlines the basic principles of non-smooth temporal substitutions and
gives exactly solvable illustrating examples. Note that the transition to nonsmooth temporal argument should be viewed as a preliminary stage of analyses.
Such stage of transformation incorporates the specifics of external loading or/and
inherent properties of a physical system into the new equations. As a result, a
much wider range of analytical and numerical methods becomes possible to apply.
This is due to the fact that the new temporal arguments vary within narrower domains and automatically capture the major temporal symmetries of the dynamics,
such periodicity and reflections. The corresponding analytical algorithms and solutions for different strongly nonlinear oscillators can be found in the reference
[2]. The typical form of such solutions is power series with respect to the triangular sine τ; see row 5 in figure 4 for explanation. Note that direct power series expansions with respect to the original time t usually make little sense for vibration
problems due to the loss of periodicity. The amplitude and frequency modulated
motions can be analyzed by adding a slow time argument to representation (10)
and then using the idea of two variables or multiple scale expansions. Let us men-
12
tion also a new area of applications, which is being developed due to an interesting observation that the temporal mode shapes of the phase variable, describing
the 1:1 resonance energy exchange between weakly nonlinear oscillators, resembles the dynamic states of impact oscillator [6]. In particular, it was found that
such “impacts” take place when the entire energy is involve into the exchange
process.
Finally, let us summarize the revealed links between specific cases of the
Euclidean rigid transformations, the induced algebraic structures, and linear and
nonlinear dynamics. The class of rigid transformations T of an arbitrary vector r
is described by the expression T ( r ) = Ar + b , where A and b are the orthogonal
matrix and a constant translation vector, respectively. Then the above mentioned
logical links are illustrated by the table, which can be viewed as an extension of
figure 4,
+ 1
⇔ Rotation
⇔ i 2 = −1 Elliptic numbers
⇔ Linear dynamics

det( A) =  0 ⇔ Singular case
⇔ i 2 = 0 Parabolic numbers
⇔ ?
− 1 ⇔ Reflection ⇔ i 2 = +1 Hyperbolic numbers ⇔ Nonlinear dynamics

where the singular case generates an open question.
References
1. Pilipchuk V.N.: Transformation of oscillating systems by means of a pair of
nonsmooth periodic functions. Dokl. Akad. Nauk Ukrain. SSR, Ser. A (4),
1988, 37–40.
2. Pilipchuk V.N.: Nonlinear dynamics: between linear and impact limits. Berlin,
Springer, 2010.
3. Kisil V.V.: Induced representations and hypercomplex numbers. Advances in
Applied Clifford Algebras, 23(2), 2013, 417-440.
4. http://en.wikipedia.org/wiki/Split-complex_number
5. Lyapunov A.M.: Investigation of a singular case of the problem of stability of
motion. Math. Sbornik, 17, 1893, 252-333.
6. Manevitch L.I.: New approach to beating phenomenon in coupled nonlinear
oscillatory chains. Arch. Appl. Mech., 77, 2007, 301-312.